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Numerical Semigroup
In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup. For example, while the set is a numerical semigroup, the set is not because 1 is in the set and 1 + 1 = 2 is not in the set. Numerical semigroups are commutative monoids and are also known as numerical monoids. The definition of numerical semigroup is intimately related to the problem of determining nonnegative integers that can be expressed in the form ''x''1''n''1 + ''x''2 ''n''2 + ... + ''x''''r'' ''n''''r'' for a given set of positive integers and for arbitrary nonnegative integers ''x''1, ''x''2, ..., ''x''''r''. This problem had been considered by several mathematicians like Frobenius (1849–1917) and Sylvester (1814–1897) at the end of the 19th century. During the second half of the twentieth c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well Temperament
Well temperament (also good temperament, circular or circulating temperament) is a type of tempered tuning described in 20th-century music theory. The term is modeled on the German word ''wohltemperiert''. This word also appears in the title of J. S. Bach's famous composition "Das wohltemperierte Klavier", ''The Well-Tempered Clavier''. Origins As used in the 17th century, the term "well tempered" meant that the twelve notes per octave of the standard keyboard were tuned in such a way that it was possible to play music in all major or minor keys that were commonly in use, without sounding perceptibly out of tune. One of the first attestations of the concept of "well tempered" is found in a treatise in German by the music theorist Andreas Werckmeister. In the subtitle of his ''Orgelprobe'', from 1681, he writes: The words and were subsequently combined into . A modern definition of "well temperament", from Herbert Kelletat, is given below: : In most tuning systems used be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup Theory
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylver Coinage
Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 of '' Winning Ways for Your Mathematical Plays''. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers. The player who cannot name such a number loses. For instance, if player A opens with 2, B can win by naming 3. Sylver coinage is named after James Joseph Sylvester, who proved that if ''a'' and ''b'' are relatively prime positive integers, then (''a'' − 1)(''b'' − 1) − 1 is the largest number that is not a sum of nonnegative multiples of ''a'' and ''b''. Thus, if ''a'' and ''b'' are the first two moves in a game of sylver coinage, this formula gives the largest number that can still be played. More generally, if the greatest common divisor of the moves played so far is ''g'', then only finitely many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Classes Of Semigroups
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ''ab'' = ''ba'' for all elements ''a'' and ''b'' in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively. In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematica Scandinavica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ..., plotting Function (mathematics), functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in ''Mathematica''. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. __TOC__ Notebook in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Number
The coin problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that asks for the largest monetary amount that cannot be obtained using only coins of specified denominations, for example, the largest amount that cannot be obtained using only coins of 3 and 5 units is 7 units. The solution to this problem for a given set of coin denominations is called the Frobenius number of the set. The Frobenius number exists as long as the set of coin denominations has no common divisor greater than 1. There is an explicit formula for the Frobenius number when there are only two different coin denominations, ''x'' and ''y'': the Frobenius number is then ''xy'' − ''x'' − ''y''. If the number of coin denominations is three or more, no explicit formula is known. However, for any fixed number of coin denominations, there is an algorithm computing the Frobenius number in polynomial time (in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedding Dimension
This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings are assumed to be commutative with identity 1. !$@ A B C D E F G H . I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup Forum
Semigroup Forum (print , electronic ) is a mathematics research journal published by Springer. The journal serves as a platform for the speedy and efficient transmission of information on current research in semigroup theory. Coverage in the journal includes: algebraic semigroups, topological semigroups, partially ordered semigroups, semigroups of measures and harmonic analysis on semigroups, transformation semigroups, and applications of semigroup theory to other disciplines such as ring theory, category theory, automata, and logic. Semigroups of operators were initially considered off-topic, but began being included in the journal in 1985. Contents Semigroup Forum features survey and research articles. It also contains research announcements, which describe new results, mostly without proofs, of full length papers appearing elsewhere as well as short notes, which detail such information as new proofs, significant generalizations of known facts, comments on unsolved problems, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the other harmonics are known as ''higher harmonics''. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a '' harmonic series''. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz. In music, harmonics are used on string instruments and wind instrum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below). Overview Definition The ''greatest common divisor'' (GCD) of two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the largest s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
